Student books : The brothers Bernoulli

ABOUT 300 years ago, the brilliant Swiss mathematician Johann Bernoulli set a
problem for “the shrewdest mathematicians of all the world”: into what shape
must a piece of wire be bent so that a frictionless bead can slide down from one
point to any other in the least possible time?

It sounds like a student’s maths problem from hell; certainly it is not all
it seems. The intuitively obvious answer—a straight line—only holds
if the two points are vertically above each other. If they are in any other
direction, a straight line will not make the most of the vertical pull of
gravity to maximise the bead’s speed along the wire, and thus minimise its
descent time.

Bernoulli’s ulterior motive in setting the problem was to trip up his older
brother, Jakob. In the event he failed, as Jakob was an even more brilliant
mathematician. But he did provoke the invention of one of the most powerful
tools in modern theoretical physics: the calculus of variations.

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Mathematically, the calculus of variations is all about finding a function
that connects two points in such a way that a certain integral has a stationary
value. This may sound terribly abstract, but it turns out to be the essence of
many problems in physics. Bernoulli’s teaser, for example, boils down to finding
a function describing the shape of the wire joining two points such that the
integral giving the total journey time is as small as possible. Variational
techniques reveal the solution to be a cycloid: the shape described by a point
on the rim of a moving wheel.

More impressively, it is possible to derive Snell’s law of optics, Newton’s
laws of motion and even Schrödinger’s equation in quantum theory from the
same basic notion of finding paths that minimise a specific integral.

This ability of variational methods to derive fundamental results in physics
hints at their being a kind of short cut towards truly fundamental insights into
the design of the cosmos. Certainly that was the way the physicist Richard
Feynman saw it: while still at school he was deeply impressed by their powers
after being shown them by his physics teacher, and he went on to show their
power in quantum theory.

Don Lemons’s short book is aimed at bringing these techniques within reach of
the average undergraduate physics student. His approach is to show
systematically how just a handful of broad principles—least time, least
potential energy, least action—can, when fed through the machinery of the
calculus of variations, reveal connections between disparate areas of
physics.

For example, the whole of ray optics follows from the idea that light always
takes the path that minimises journey time. Similarly, least potential energy
explains a host of mechanical phenomena, from the shape of hanging overhead
power cables to the behaviour of loaded beams. The concept of least
action—that is, minimal difference between kinetic and potential
energy—explains why a ball follows a parabola when it is thrown, and why
heavenly bodies orbit in conic sections.

Lemons includes examples and exercises on these and many more, providing an
inspiringly broad-ranging introduction to the impressive unifying power of
variational methods. He also provides plenty of historical asides, which show
how the principles emerged from quasi-philosophical beliefs in the “perfection”
of nature.

If I have a criticism of the book it is that it might have said more about
the limitations of variational methods: they can’t be used in dissipative
systems, such as those involving frictional forces. On a different level, it
could have said more about the mysterious efficacy of variational principles in
fundamental physics. Just why does the Universe obey such principles? And is
there an “ultimate” principle from which least time, least action and the others
can themselves be derived?

Such questions are the stuff of future Nobel prizes. Lemons provides an
accessible introduction to their origin.